Split Metacyclic p-Groups That Are A-E Groups

Abstract

A group G is an A-E group if the endomorphism nearring of G generated by its automorphisms equals the endomorphism nearring generated by its endomorphisms. In this paper we set out to determine those p-groups G that are semidirect products of cyclic groups and are A-E groups. We show that no such groups exist when p = 2. When p is odd, we show that G is an A-E group whenever the nilpotency class of G is less than p. Examples are given to show no conclusion can be drawn when the nilpotency class is greater than or equal to p.